Math Is Fun Forum

Kurre

Member

Registered: 2006-07-18

Posts: 280

prime number problem

Let p be a prime. Prove that if q|2^p-1, then p|q-1.

Please dont give full answer, only hints.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: prime number problem

Consider the multiplicative group

of nonzero integers

. It is clear that

must be odd; thus, by Fermat’s little theorem, the order of 2 in the group must divide

. If

then the order of 2 also divides

and hence is

since

is prime.

Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: prime number problem

thanks

whatismath

Member

Registered: 2007-04-10

Posts: 19

Re: prime number problem

I think we do not need to assume

to be prime.
Actually, Jane's argument shows

for all prime factor

of

.
Any factor

of

is a product of
such

, i.e.

for some

.
Hence

,
or

.

Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: prime number problem

I did not write that q must be prime